Introduction
Local cohomology of modules was introduced by A. Grothendieck. Publication of R. Hartshorne’s lecture notes of Grothendieck’s Harvard seminars established the importance of local cohomology as an effective tool to solve many problems in commutative algebra and algebraic geometry. This workshop will introduce basics and advanced aspects of this theory along with major applications and recent developments. The speakers will develop necessary background in commutative algebra to train the participants in the first week so that they can follow advanced lectures in the second week. Some of the participants will also present their works on local cohomology.

Course Plan in Brief

1.  Manoj Kummini: Injective modules : Injective Modules, existence of injective hull, structure theorem for irreducible injective modules, examples of injective modules, Matlis duality.
2. Tony Puthenpurakal: Basics of local cohomology : Resolutions and derived functors, Local Cohomology functor, direct limit of Ext modules, direct limit of Koszul cohomology modules., depth, nonvanishing and vanishing theorems.
3.  Dilip Patil : Cohen-Macaulay rings : Koszul complex, regular sequences, characterisation of depth via Ext modules,, Cohen-Macaulay modules and their characterisations, unmixedness theorems, examples of CM rings, Auslander-Buchsbaum formula.
4.  H. Ananthnarayan: Gorenstein rings : various definitions of Gorenstein rings, Bass numbers, injective resolutions, canonical module, local duality, examples of Gorenstein rings, Serre’s theorem.
5.  Krishna Hanumanthu: Serr’e FAC paper : Sheaves, Cech cohomology, Flasque sheaves, relation between local cohomology and sheaf cohomology, Grothendieck-Serre difference formula for Hilbert functions, projective Varieties: graded local cohomology, sheaves on projective varieties, global sections and cohomology.
6.  Markus Brodmann: D-modules :  Filtered algebras, derivations,Weyl algebra, the standard basis, weighted degrees and filtrations, weighted associated graded rings, filtered modules, D-modules, Gr ̈obner bases, weighted orderings, standard degree and Hilbert polynomials.
7.  G. Lyubeznik: Characteristic p methods for local cohomology : Frobenius action on local cohomology, F -rings, parafactoriality, Samuel’s conjecture via local cohomology, Cohen-Macaulay algebras: the Huneke-Lyubeznik theorem.
8.  N.T. Cuong: Sequentially Cohen-Macaulay modules : Filtrations and good systems of parameters, sequentially Cohen-Macaulay modules, characterisation of sequentially Cohen-Macaulay modules by good systems of parameters.
9.  Le Tuan Hoa: Castelnuovo-Mumford regularity : Characterizations of Castelnuovo-Mumford regularity via local cohomology, Mumford’s characterization of regularity of sheaves on projective space and its extension by Eisenbud-Goto, regularity and reduction numbers of large powers of ideals, bounding regularity in terms of degrees of defining equations, regularity and Hilbert coefficients, complexity of Gr ̈obner bases, some open problems.
10. Peter Schenzel: Bezout’s Theorem and local cohomology : Rees and form rings, Hilbert function, Hilbert polynomial, multiplicities, Auslander-Buchsbaum-Serre formula for multiplicities of parameter ideals, subcomplexes of Koszul and Cech-complexes, inequalities for local intersection numbers in Bezout’s, theorem, corrections of results of Pritchard and Greuel-Lossen-Justin

  Schedule of Lectures and Tutorials